Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions

We have investigated the next-to-next-to-leading order (NNLO) corrections to inclusive hadron production in e+e− annihilation and the related parton fragmentation distributions, the ‘time-like’ counterparts of the ‘space-like’ deep-inelastic structure functions and parton densities. We have re-derived the corresponding second-order coefficient functions in massless perturbative QCD, which so far had been calculated only by one group. Moreover we present, for the first time, the third-order splitting functions governing the NNLO evolution of flavour non-singlet fragmentation distributions. These results have been obtained by two independent methods relating time-like quantities to calculations performed in deep-inelastic scattering. We briefly illustrate the numerical size of the NNLO corrections, and make a prediction for the difference of the yet unknown timelike and space-like splitting functions at the fourth order in the strong coupling constant. In this letter we address the evolution of the parton fragmentation distributions Dh and the corresponding fragmentation functions Fh a in e +e− annihilation, e+e− → γ, Z → h+X where 1 σtot d2σ dx dcosθ = 3 8 (1+ cos2 θ) Fh T + 3 4 sin2 θ F h L + 3 4 cosθ Fh A . (1) Here θ represents the angle (in the center-of-mass frame) between the incoming electron beam and the hadron h observed with four-momentum p, and the scaling variable reads x = 2pq/Q2 where q with q2 ≡ Q2 > 0 is the momentum of the virtual gauge boson. The transverse (T ), longitudinal (L) and asymmetric (A) fragmentation functions in Eq. (1) have been measured especially at LEP, see Ref. [1] for a general overview. Disregarding corrections suppressed by inverse powers of Q2, these observables are related to the universal fragmentation distributions Dh by F a (x,Q ) = ∑ f=q, q̄,g ∫ 1 x dz z ca,f ( z,αs(Q) ) D f (x z ,Q ) . (2) The coefficient functions ca,f in Eq. (2) have been calculated by Rijken and van Neerven in Refs. [2–4] up to the next-to-next-to-leading order (NNLO) for Eq. (1), i.e., the second order in the strong coupling as ≡ αs(Q)/(4π). Below we will present the results of a re-calculation of these functions by two approaches differing from that employed in Refs. [2–4]. Besides the second-order coefficient functions, a complete NNLO description also requires the third-order contributions to the splitting functions (so far calculated only up to the second order [5–7]) governing the scale dependence (evolution) of the parton fragmentation distributions. In a notation covering both the (time-like q, σ = 1) fragmentation distributions and the (spacelike q, Q2 ≡−q2, σ = −1) parton distributions, the flavour non-singlet evolution equations read d d lnQ2 f ns σ (x,Q 2) = ∫ 1 x dz z Pns σ ( z,αs(Q) ) f ns σ (x z ,Q2 ) (3) with Pns σ ( x,αs(Q) ) = as P (0)ns(x) + a2 s P (1)ns σ (x) + a 3 s P (2)ns σ (x) + . . . . (4) The superscript ‘ns’ in Eqs. (3) and (4) stands for any of the following three types of combinations of (parton or fragmentation) quark distributions, f ± ik = qi ± q̄i − (qk ± q̄k) , f v = ∑ nf r=1(qr − q̄r) , (5) where nf denotes the number of active (effectively massless) flavours. As detailed below, we have obtained the so far unknown time-like NNLO splitting functions P σ=1 (x) in Eq. (4). As already indicated in Eq. (4), the space-like and time-like non-singlet splitting functions are identical at the leading order (LO) [8], a fact known as the Gribov-Lipatov relation. This relation does not hold beyond LO in the usual MS scheme adopted also in this letter. However, the space-like and time-like cases are related by an analytic continuation in x, as shown in detailed diagrammatic analyses [5,9] at order αs , see also Refs. [10,11]. Moreover, another approach relating the non-singlet splitting functions has been proposed in Ref. [12]. Hence it should be possible to derive time-like quantities from the space-like results computed to order α3 s in Refs. [13–15].